{"paper":{"title":"Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Ahmad Khalkhuzhaev, Mardon Pardabaev, Shokhrukh Yu. Kholmatov","submitted_at":"2019-10-03T09:18:31Z","abstract_excerpt":"We consider the family $\\hat h_\\mu:=\\hat\\varDelta\\hat \\varDelta - \\mu \\hat v,$ $\\mu\\in\\mathbb{R}, $ of discrete Schr\\\"odinger-type operators in $d$-dimensional lattice $\\mathbb{Z}^d$, where $\\hat \\varDelta$ is the discrete Laplacian and $\\hat v$ is of rank-one. We prove that there exist coupling constant thresholds $\\mu_o,\\mu^o\\ge0$ such that for any $\\mu\\in[-\\mu^o,\\mu_o]$ the discrete spectrum of $\\hat h_\\mu$ is empty and for any $\\mu\\in \\mathbb{R}\\setminus[-\\mu^o,\\mu_o]$ the discrete spectrum of $\\hat h_\\mu$ is a singleton $\\{e(\\mu)\\},$ and $e(\\mu)<0$ for $\\mu>\\mu_o$ and $e(\\mu)>4d^2$ for $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.01369","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1910.01369/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}